Wandering continua in Julia sets of rational maps: are arcs and Jordan curves the only possible types?

A wandering continuum of a rational map $f$ is a non-degenerate continuum $K$ contained in the Julia set $J(f)$ whose forward images are pairwise disjoint. Such objects exist — Sullivan's no-wandering-domains theorem has no counterpart on the Julia-set side — but every example known to date is topologically one of two types: an arc or a Jordan curve. McMullen's singularly perturbed maps have Cantor families of wandering Jordan curves; for post-critically finite rational maps, Cui, Peng and Tan proved that a non-full (separating) wandering continuum exists exactly when the map has a Cantor multicurve, and is then eventually a Jordan curve; Cui and Gao proved that a Lattès map admits a full wandering continuum if and only if it is flexible, the full ones being line segments in flat geodesics — arcs — while flexible Lattès maps also carry wandering Jordan curves; and wandering arcs are recorded in Cremer quadratic Julia sets. This record states the problem as posed by Evdoridou, Martí-Pete and Rempe in 2026: does there exist a rational map whose Julia set contains a non-degenerate wandering continuum that is neither an arc nor a Jordan curve, and more generally, which topological types of continua can wander in a rational Julia set?

Background

The iteration of a rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ of degree at least two splits the Riemann sphere into the Fatou set, on which the family of iterates $\{f^{n}\}$ is normal, and the Julia set $J(f)$, its complement, where the dynamics is chaotic. The no-wandering-domains theorem states that every Fatou component of a rational map is eventually periodic; on the Julia-set side, however, compact connected sets can genuinely wander, and how complicated they can be is not known.

By a continuum we mean a compact connected set with more than one point. A continuum $K\subset J(f)$ is a wandering continuum if $f^{n}(K)\cap f^{m}(K)=\emptyset$ for all $n>m\ge 0$. Single points with infinite forward orbits are degenerate wandering continua and always exist; the interesting objects are the non-degenerate ones, and the basic question about them is topological: an arc is a continuum homeomorphic to $[0,1]$, a Jordan curve one homeomorphic to the circle $S^{1}$, and one asks whether anything else can wander.

For polynomials, external-ray and lamination methods — external rays being the dynamically distinguished curves in the basin of infinity that land on the Julia set, and laminations the combinatorial objects recording which of them land at a common point — give a nearly complete dichotomy. If the Julia set is disconnected, all wandering components of the filled Julia set are points. If the Julia set is connected and the map has no irrational indifferent cycles (a periodic point is irrationally indifferent when its multiplier is $e^{2\pi i\theta}$ with irrational $\theta$; it is a Siegel point if the map is locally linearizable there and a Cremer point otherwise), then there are no wandering continua exactly when the Julia set is locally connected, i.e. when each point admits arbitrarily small connected neighborhoods in the set — a theorem recorded, with references, by Cui–Gao and by Cui–Peng–Tan, with origins in work of Blokh–Levin and Kiwi. Zeng gave a new proof of the 'locally connected implies none' direction: his Corollary 1.3 shows that a polynomial of degree $d\ge 2$ with locally connected Julia set admits no wandering continuum. Wandering branch points — points at which the Julia set locally branches into three or more arcwise directions — do exist in locally connected cubic dendrite Julia sets (a dendrite is a locally connected continuum containing no Jordan curve; reproved constructively by Buff, Canela and Roesch), but a branch point is a point, a degenerate wandering object, not a continuum. Evdoridou, Martí-Pete and Rempe further record, attributing it to Cheraghi, that the Julia sets of some quadratic polynomials with non-linearisable irrationally indifferent fixed points contain wandering arcs.

For non-polynomial rational maps the known classifications are partial. McMullen proved that for $f_{\lambda}(z)=z^{2}+\lambda/z^{3}$ with $\lambda>0$ small, the Julia set is a Cantor set of Jordan curves, uncountably many of which wander; for geometrically finite rational maps all wandering Julia components are points or Jordan curves (as recorded by Cui and Gao). Cui, Peng and Tan introduced Cantor multicurves for the post-critically finite case (a map is post-critically finite, pcf, when the forward orbits of its critical points form a finite set; a multicurve is a finite family of simple closed curves in the complement of the postcritical set, pairwise non-homotopic and non-contractible there, and a Cantor multicurve is one whose pullbacks under iteration keep generating new homotopy classes). They proved that for a pcf rational map a wandering continuum that is not full — full meaning having connected complement — exists if and only if the map has a Cantor multicurve, and that then the continuum is eventually a Jordan curve. The complementary case of full wandering continua they left open, conjecturing that among pcf maps only flexible Lattès maps admit them. A Lattès map is a rational map covered by an affine self-map of a complex torus; the flexible ones are precisely those admitting non-trivial quasiconformal deformations (quasiconformal maps are homeomorphisms of the sphere that distort angles only in a bounded fashion). Cui and Gao settled the Lattès case of this conjecture: a Lattès map admits an always-full wandering continuum — one $K$ for which every iterate $f^{n}(K)$, $n\ge 0$, is itself full — if and only if it is flexible, every always-full wandering continuum of a torus covering being a line segment in a flat geodesic — an arc; moreover, a Lattès map is flexible exactly when it has a Cantor multicurve, so flexible Lattès maps also admit wandering Jordan curves, while rigid Lattès maps have no non-degenerate wandering continua at all. Thus, across all rational maps, every wandering continuum currently known is an arc or a Jordan curve.

The contrast with transcendental dynamics suggests the answer is not obviously negative: Rempe constructed a transcendental entire function whose Julia set contains, for every arc-like continuum $X$ (a continuum expressible as an inverse limit of arcs), a wandering continuum homeomorphic to $X$, and later work realizes essentially arbitrary compact sets as wandering sets of transcendental functions. No analogous construction, and no obstruction, is known for rational maps. Blokh, Curry and Oversteegen's finest monotone models and Kiwi's laminations, the main tools for describing continua in polynomial Julia sets, collapse continua rather than classify them, which is one reason the topological type of the wandering continua that exist in non-locally-connected polynomial Julia sets has resisted classification.

Problem Statement

Does there exist a rational map $f$ of degree at least two whose Julia set contains a non-degenerate wandering continuum — a continuum $K\subset J(f)$ with $f^{n}(K)\cap f^{m}(K)=\emptyset$ for all $n\neq m$ — that is neither an arc (homeomorphic to $[0,1]$) nor a Jordan curve (homeomorphic to $S^{1}$)?

More generally, classify the topological types of non-degenerate wandering continua that can occur in Julia sets of rational maps: determine whether arcs and Jordan curves are the only possibilities.

A positive answer exhibits a rational map of degree at least two together with a non-degenerate continuum in its Julia set whose forward images are pairwise disjoint and which is topologically neither an arc nor a Jordan curve. A negative answer is a theorem covering all rational maps of degree at least two — polynomials and non-polynomials, connected and disconnected Julia sets, including maps with irrational indifferent (Siegel or Cremer) cycles. These are two facets of a single classification objective, not independent problems: the existence question is the smallest open instance of the classification program.

References

1. V. Evdoridou, D. Martí-Pete, L. Rempe (2026), 'Wandering dynamics of transcendental functions', preprint, arXiv:2602.09952, https://arxiv.org/abs/2602.09952
2. J. Zeng (2020), 'Criterion for rays landing together', Transactions of the American Mathematical Society 373, 6479-6502, DOI 10.1090/tran/8088, arXiv:1503.05931, https://arxiv.org/abs/1503.05931
3. A. Blokh, C. Curry, L. Oversteegen (2011), 'Locally connected models for Julia sets', Advances in Mathematics 226, 1621-1661, DOI 10.1016/j.aim.2010.08.011, arXiv:0809.3754, https://arxiv.org/abs/0809.3754
4. G. Cui, W. Peng, L. Tan (2016), 'Renormalizations and wandering Jordan curves of rational maps', Communications in Mathematical Physics 344, 67-115, DOI 10.1007/s00220-016-2623-x, arXiv:1403.5024, https://arxiv.org/abs/1403.5024
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